Is the Square Root of 90 a Rational Number? Discover the Truth!

To determine whether the square root of 90 is a rational number, we need to understand the definitions of rational and irrational numbers.

Definition of Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. In other words, a number r is rational if it can be written in the form r = p/q, where p and q are integers and q ≠ 0.

Definition of Irrational Numbers

An irrational number is a number that cannot be expressed as a simple fraction - its decimal form is non-repeating and non-terminating.

Analysis of the Square Root of 90

Let's analyze the square root of 90 to see if it can be expressed as a fraction of two integers.

The prime factorization of 90 is:

• 90 = 2 × 3² × 5

Since 90 is not a perfect square (i.e., it cannot be expressed as the square of an integer), its square root will not be an integer.

We can represent the square root of 90 as:

\(\sqrt{90}\)

This can be further simplified to:

\(\sqrt{90} = \sqrt{2 \times 3^2 \times 5} = 3 \times \sqrt{10}\)

Now, we observe that \(\sqrt{10}\) is not a rational number, as 10 is not a perfect square.

Conclusion

Since \(\sqrt{10}\) is irrational, \(3 \times \sqrt{10}\) is also irrational. Therefore, \(\sqrt{90}\) is an irrational number.

In summary, the square root of 90 is not a rational number. It cannot be expressed as a fraction of two integers, and its decimal expansion is non-repeating and non-terminating.

The question of whether the square root of 90 is a rational number is intriguing. Rational numbers are numbers that can be expressed as the quotient of two integers. To determine if the square root of 90 fits this definition, we must delve into its properties and mathematical characteristics. This article will guide you through the process step by step, providing clarity on this mathematical inquiry.

• We will begin by understanding the definitions of rational and irrational numbers.
• Next, we will explore the prime factorization method to break down the number 90.
• We'll then discuss perfect squares and their square roots to provide context.
• The process of simplifying the square root of 90 will be explained in detail.
• We'll express the square root of 90 as a product to further analyze its nature.
• The decimal expansion of the square root of 90 will be examined.
• Finally, we'll present a proof to determine if the square root of 90 is irrational.

Through this comprehensive exploration, you'll gain a thorough understanding of whether the square root of 90 is a rational number and why.

The prime factorization method is a powerful tool for breaking down a number into its basic building blocks, which are prime numbers. This method is particularly useful in simplifying square roots and understanding the properties of numbers. Let's apply the prime factorization method to the number 90.

1. Identify Prime Factors: Start by dividing 90 by the smallest prime number, 2.
   • \( 90 \div 2 = 45 \)
   • Since 45 is not divisible by 2, move to the next smallest prime number, 3.
   • \( 45 \div 3 = 15 \)
   • \( 15 \div 3 = 5 \)
   • 5 is a prime number.
2. Write the Prime Factorization: The prime factorization of 90 is:

   \[ 90 = 2 \times 3^2 \times 5 \]

3. Use the Prime Factorization for Simplification: Knowing the prime factorization allows us to simplify the square root of 90:

   \[ \sqrt{90} = \sqrt{2 \times 3^2 \times 5} \]

   We can simplify this further by separating the perfect square from the other factors:

   \[ \sqrt{90} = \sqrt{3^2 \times 2 \times 5} = 3 \sqrt{10} \]

Through the prime factorization method, we see that the square root of 90 simplifies to \( 3 \sqrt{10} \), which is not a rational number because \( \sqrt{10} \) is an irrational number. This method not only helps in simplifying square roots but also provides insight into the nature of the number in question.

A perfect square is an integer that can be expressed as the square of another integer. For example, 9 is a perfect square because \( 9 = 3^2 \).

The square root of a number \( x \), denoted as \( \sqrt{x} \), is a value that, when multiplied by itself, gives \( x \). If \( \sqrt{x} \) is an integer, then \( x \) is a perfect square.

Now, let's determine if \( \sqrt{90} \) is a rational number:

1. Prime Factorization of 90:

   Start by finding the prime factors of 90:
   \[
   90 = 2 \times 3^2 \times 5
   \]

2. Identify Perfect Squares:

   Look at the prime factorization to identify pairs of factors that are perfect squares:
   \[
   90 = 2 \times 3^2 \times 5
   \]
   Here, \( 3^2 \) is a perfect square.

3. Expression of \( \sqrt{90} \):

   Express \( \sqrt{90} \) in terms of its prime factors:
   \[
   \sqrt{90} = \sqrt{2 \times 3^2 \times 5} = 3\sqrt{10}
   \]

4. Rational or Irrational:

   Since \( \sqrt{10} \) is irrational (as proved in other sections), \( \sqrt{90} = 3\sqrt{10} \) is also irrational.

Therefore, the square root of 90, \( \sqrt{90} \), is irrational because it cannot be expressed as a fraction of two integers.

To simplify the square root of 90, we can use the prime factorization method. This involves expressing 90 as a product of its prime factors:

\[ 90 = 2 \times 3^2 \times 5 \]

We can then use these factors to simplify the square root:

\[ \sqrt{90} = \sqrt{2 \times 3^2 \times 5} \]

Since the square root of a product is the product of the square roots, we can simplify this further:

\[ \sqrt{90} = \sqrt{2} \times \sqrt{3^2} \times \sqrt{5} \]

Recognizing that the square root of \(3^2\) is 3, we get:

\[ \sqrt{90} = \sqrt{2} \times 3 \times \sqrt{5} \]

So, the simplified form is:

\[ \sqrt{90} = 3\sqrt{10} \]

Decimal Form

We can also express the square root of 90 in its decimal form. By using a calculator or mathematical software, we find:

\[ \sqrt{90} \approx 9.4868 \]

This indicates that the square root of 90 is not a perfect square, as its decimal representation is non-terminating and non-repeating, confirming it is an irrational number.

To express the square root of 90 as a product of its factors, we can use the prime factorization method. First, we find the prime factors of 90:

• 90 = 2 × 3 × 3 × 5

We can group the factors to simplify the square root:

\[
\sqrt{90} = \sqrt{2 \times 3^2 \times 5}
\]

Next, we simplify the expression by taking the square root of the perfect square (which is 3^2) out of the radical:

\[
\sqrt{90} = \sqrt{2 \times 3^2 \times 5} = 3 \times \sqrt{2 \times 5} = 3 \sqrt{10}
\]

Therefore, the square root of 90 can be expressed as a product:

\[
\sqrt{90} = 3 \sqrt{10}
\]

The decimal expansion of the square root of 90 is a non-repeating and non-terminating decimal, indicating that it is an irrational number. The principal square root of 90 is approximately 9.486832980505138.

To understand the decimal expansion, let's examine the value step-by-step:

• The square root of 90 can be expressed as \( \sqrt{90} \).
• Using a calculator or a mathematical method, \( \sqrt{90} \) is approximately 9.486832980505138.
• This value can be verified using various numerical methods such as the Newton-Raphson method or long division.

For practical purposes, the square root of 90 is often rounded to a certain number of decimal places:

• To 2 decimal places: 9.49
• To 3 decimal places: 9.487

Here is the value of \( \sqrt{90} \) rounded to several decimal places:

Due to the irrational nature of \( \sqrt{90} \), its decimal expansion continues infinitely without repeating.

The value of \( \sqrt{90} \) is crucial in various mathematical applications and problems where precise calculations are required.

To prove that the square root of 90 is irrational, we will use a method that involves the prime factorization of 90. An irrational number cannot be expressed as a ratio of two integers, and its decimal expansion is non-repeating and non-terminating.

1. 
   Assume the contrary: Suppose that √90 is rational. This means we can write it as a fraction of two integers, say \( \frac{a}{b} \), where \( a \) and \( b \) have no common factors other than 1, and \( b \neq 0 \).

2. 
   Square both sides: \( \sqrt{90} = \frac{a}{b} \implies 90 = \frac{a^2}{b^2} \implies a^2 = 90b^2 \).

3. 
   Prime factorization: The prime factorization of 90 is \( 2 \times 3^2 \times 5 \). Since \( a^2 = 90b^2 \), the prime factors of \( a^2 \) must match the prime factors of \( 90b^2 \).

   
   However, \( 90b^2 \) includes the prime factor 5, and for \( a^2 \) to match \( 90b^2 \), \( a \) must also include the prime factor 5. This means \( a \) must be divisible by 5.

4. 
   Contradiction: If \( a \) is divisible by 5, then we can write \( a = 5k \) for some integer \( k \). Substituting back, we get \( (5k)^2 = 90b^2 \implies 25k^2 = 90b^2 \implies 5k^2 = 18b^2 \).

   
   This implies \( 5k^2 = 18b^2 \), meaning 18 must also be divisible by 5, which is a contradiction since 18 and 5 have no common factors.

5. 
   Conclusion: The assumption that √90 is rational leads to a contradiction. Therefore, √90 is irrational.

Irrational numbers, despite their abstract nature, have numerous practical applications in various fields. Here are some significant uses:

• Finance: Irrational numbers are crucial in financial calculations, particularly in the computation of compound interest. The sum of infinite series involving irrational numbers is often used to determine interest rates and investment growth.
• Engineering and Design: The constant \( e \) (Euler's number), an irrational number, is widely used in engineering for calculating growth and decay rates, in signal processing, and in the design of various systems. It helps in modeling real-world scenarios that involve continuous growth or decay.
• Construction: In construction and architecture, irrational numbers like \(\pi\) are essential for designing and constructing cylindrical and spherical structures. The calculation of areas and circumferences of circles relies on \(\pi\), which is an irrational number.
• Nature and Art: Irrational numbers appear in nature, such as in the patterns of leaves, the arrangement of seeds in a sunflower, and other natural spirals. The golden ratio, another irrational number, is used in art and architecture to achieve aesthetically pleasing proportions.
• Scientific Research: Irrational numbers are used in various scientific calculations, including those involving waves, quantum mechanics, and relativity. They help in accurately describing the natural phenomena that cannot be expressed as exact fractions.

These examples illustrate how irrational numbers, though non-terminating and non-repeating, play a critical role in practical applications, influencing many aspects of modern life and technology.